Mathematical Research Letters

Volume 9, Issue 3, March 2002 pp. 363-378.

The Pressure Function for Products of Non-negative Matrices

Authors: De-Jun Feng and Ka-Sing Lau
Author institution: Tsinghua University, and The Chinese University of Hong Kong

Summary: Let $(\Sigma_A, \sigma)$ be a subshift of finite type and let $M(x)$ be a continuous function on $\Sigma_A$ taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where $M(x)$ takes finite values $M_1, \cdots, M_m$. The pressure function reduces to $P(q):=\lim_{n\rightarrow \infty}\frac{1}{n} \log \sum_{\hskip -2pt J \in \Sigma_{A, n}} \|M_J\|^q$. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps.

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